The Probability Pipe Organ

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Runs:

Flips/run:

This page demonstrates how experiments with random data converge
toward the predictions of probability theory as more and more
experiments are run.

The panel above initially shows the bell-curve normal
distribution approximating the binomial distribution for
an experiment consisting of 16 coin flips. To see the bell curve
for experiments with different numbers of flips, enter a value between
4 and 1024 inclusive in the “Flips/run” box before
pressing “Start”. For large numbers of flips, the bell curve
will be very narrow, since the probability of a large excess of heads
or tails is very low. In every case, the peak of the bell curve, the
most probable result, represents an equal numbers of heads and tails.

Pressing the “Start” button begins a series of simulated
experiments, each consisting of the number of flips specified by
“Flips/run”. The number of heads are tallied and
displayed as a histogram superimposed on the normal curve. As more
and more experiments are run, the scale of the histogram bars is
adjusted so the tallest bar remains as high as the peak in the normal
curve. Press “Stop” to suspend the running of
experiments; press “Run” to resume after a pause.
“Reset” stops the experiment if running and clears the
result for a new run. If you like, you can enter a new value for
“Flips/run” whenever the experiment is stopped; previous
results will be cleared. The “Step” button runs one
experiment and updates the histogram each time it is pressed.

Observe how at the outset, especially for experiments with relatively
few flips, results may appear to depart substantially from the chance
expectation, but as more and more experiments are run, the results
converge ever more closely on the prediction from probability theory.
The initial outlying results “scroll down” as more and more
experiments produce the most probable outcomes.